All of the 5th grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$4.00$ each for students, and the group paid $$49.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$17.00$ each for teachers and $$10.50$ each for students, and the group paid $$118.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4y = 49}$ ${17x+10.5y = 118}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-17x-8y = -98}$ ${17x+10.5y = 118}$ Add the top and bottom equations together. $ 2.5y = 20 $ $ y = \dfrac{20}{2.5}$ ${y = 8}$ Now that you know ${y = 8}$ , plug it back into $ {8.5x+4y = 49}$ to find $x$ ${8.5x + 4}{(8)}{= 49}$ $8.5x+32 = 49$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 8}$ into $ {17x+10.5y = 118}$ and get the same answer for $x$ ${17x + 10.5}{(8)}{= 118}$ ${x = 2}$ There were $2$ teachers and $8$ students on the field trips.